Language mirrors the world. Every time we make a statement in a natural language like English – that there are flowers on the table, that I feel happy, that two threes are six – our statement mirrors the world, represents the world. We say it models the world, in the sense that it builds a model or representation of a corner of the world.
Change is something that happens in the world. I will be looking at how change is modelled by us all as we speak everyday English, and how it is modelled more formally in the more specialised language of disciplines like philosophy, mathematics, and computing. By comparing the informal with the formal, we will see how extraordinarily powerful and flexible everyday English is when it expresses change.
I start with the concept of a variable, which is the simplest model of change. I move on to more complex models. Finally I bring together all the models I have considered, with a quote from a news article.
What is the most elementary kind of thing that can be said to change? Mathematicians call it a variable. Intuitively, we can just think of a variable as a number that can change.
That intuitive understanding is not robust enough for a formal definition. Those who attempt to formalise mathematics – mathematical logicians, or mathematical philosophers – would protest that, of course, a number cannot change. They would try instead to define a variable as something like a symbol or name that can represent any number.
Computer scientists talk about variables, too. Traditionally, they picture a variable as a store, like a pigeon‐hole, that can store one number at a time. In practice, their concept of a variable and the mathematician’s concept of a variable are very similar, and nothing in this essay hinges on the distinctions between them.
The value of a variable is the number that it is for the time being (intuitive definition), or the number that it represents for the time being (mathematical definition), or the number that it stores for the time being (computing definition).
Where I talk of a number, I ought to be more general. I ought to include other quantities (as mathematicians call them), or data types (as computer scientists call them). These include numbers of various sorts, truth values, text strings, dates and times, temperatures, colours, arrays: you name them. But mostly I’ll talk of numbers.
What does it mean for a variable to change? A variable changes if it has one value at one time and a different value at a later time. That’s obvious. But it does show that the concept of change depends on the concept of difference and the concept of time.
Variable or value
Does the variable change, or does its value change? Ah. We can tie ourselves in knots puzzling over this one. Here is one possible train of thought. “It’s the variable that changes, obviously. Well, I don’t mean that the variable itself has changed, of course. It’s still the same variable. It has just changed its value. So it’s the value that has changed. But the value is a number, and a number can’t change!”
To manage that difficulty robustly we have to say that the variable has changed as to its value, or some such thing. But life is too short. We talk of the variable changing, and we talk of its value changing. It’s just that, while we wear our pedant’s hat, we have to remember what we mean. The more the variable changes as to its value, the more clearly we can see that it remains the same variable. Plus ça change, plus c’est la même chose.
It seems to me that natural language is wonderfully adapted to filter out that difficulty. If a friend says, “I’m going to the greenhouse to read the temperature,” we don’t say to him, “Now wait a minute: when you say ‘the temperature’ are you talking about the variable representing the temperature, or its value?” And when he reports, “The temperature has fallen,” we know exactly what he means. We don’t need to decode it into “The temperature variable has decreased as to its value.” So, not for the only time in this essay, I find myself asking, Which is the greater wisdom, the learned wisdom of the philosopher of language, or the folk wisdom inherent in the natural language of every man, woman and child?
This elegance of natural language comes at a price, however. The price is that we must beware of trains of thought like, “The temperature has fallen; the temperature is 19 degrees; therefore 19 degrees has fallen.” The fallacy there, of course, is that “the temperature” refers to the variable in the first premise, but the value in the second premise.
The concept of a variable that changes is powerful and general. We might therefore be tempted to suppose that anything that may be said to change can be regarded as a variable. But we should be wrong. We may talk naturally of a law changing, or a school changing, or a person changing, and these are not variables representing quantities or data types. So what is going on?
Laws and schools and persons are examples of what philosophers call continuants. A continuant is an entity whose identity continues to be recognisable over some extended interval of time. I will follow computer terminology and call it an object. It may be a book, or a herd of elephants, or a belief, or a semibreve. In this sense, a person too is an object: if you feel that the noun object is incompatible with personhood then you will need to read “object” as “continuant” in that context. Modern computer languages are said to be object‐oriented, because they allow the programmer to program in terms of objects and not just variables. Object‐oriented computer programming was developed not just because it makes programming easier, but because it provides for better modelling of our thinking and speaking about the world, and through that, one hopes, better modelling of the world itself. So, as we study the language of change, let us borrow some of the concepts of object‐oriented programming and become, for the moment, object‐oriented philosophers of language.
Objects have properties. To take a simple example, a plastic ball might be modelled as an object that has a Radius property and a Colour property. Those properties take values that are numbers or other data types, just as variables do. And a property may change as to its value, just as a variable may. So we may say, in variable‐oriented terms, “The colour of the ball has changed as to its value”; or, in object‐oriented terms, “The ball has changed as to the value of its Colour property”; or, in natural English, “The ball has changed colour.”
You may be wondering: Why are we spending so much time talking about how a plastic ball might be modelled as an object with properties? Would it not be simpler to follow the philosophers, call it as it is, and say directly that the ball is an object with properties? But if we think like that, we are deluding ourselves. The moment we use language to say what a plastic ball is, we are modelling it. And the moment we use a word like property in saying what a plastic ball is, we are modelling it at really quite a deep and abstract level. As a matter of fact, the modelling of the computer programmer here is more straightforward and less abstract than the modelling of the philosopher. A computer scientist can say plainly, on his own terms, what kind of thing a property is: it’s a programming construct. Ask a philosopher what kind of thing a property is, and he’ll tell you it’s a universal. But universals are highly abstract entities – much more abstract than programming constructs – and philosophers are deeply divided as to what they are.
We have seen that “The ball has changed colour” means that the ball has changed as to the value of its colour property. More generally, when we talk about an object changing, we usually mean that it has changed as to the value of one or more of its properties. We may say that our school has changed if its examination pass rate has changed, or if it has changed the age range of the pupils that it accepts, or the range of subjects that it offers, or its ethos. We could model the situation with a School object that has a PassRate property (whose data type will be a number of some sort), an AgeRange property (probably a pair of numbers, minimum and maximum), a Subjects property (perhaps an array of text strings), and an Ethos property (some more complex data type).
We saw earlier that when a variable changes (as to its value) it remains the same variable. In the same way, when an object changes (as to the values of its properties) it remains the same object.
A variable is itself a continuant in philosophical terms, and so it too could be regarded as an object: an object of a particularly simple kind, with only one property, namely its value. But that is rather trivial. A computer programmer, for example, would not think of abusing the full power (and incurring the overheads) of object‐oriented programming merely to deal with a variable, which, to the programmer, is an altogether more elementary kind of thing than an object is. Likewise a property of an object is a continuant and could be trivially be regarded as an object in its own right.
Properties that represent objects
Not all properties represent data. We can also have a property that represents an object. For example, when we model our school as an object, as well as giving it PassRate, AgeRange, Subjects and Ethos properties, we might also give it a Headmaster property, which represents not data of any data type, but an object, namely the man who is headmaster. That object will itself have properties: for example, a Name property and an Age property. If it is of importance to our school to know whether the headmaster has a beard, we might also build into our model a Beard property (taking the value True or False) for the Headmaster object. In that sort of way, all sorts of systems can be modelled using hierarchies of objects.
Change and replacement
Let’s stay with our school. Yesterday, the headmaster of the school had a beard. Today, the headmaster of the school has no beard. What has happened?
Well, either of two processes may have happened:
- Our headmaster has shaved off (or otherwise lost) his beard;
- A new headmaster has been appointed, and he has no beard.
In our object model, these two processes are represented as follows:
- Our headmaster has changed as to the value of his Beard property;
- Our school has changed as to the value of its Headmaster property.
How can we be sure there is no third possibility? Well, look at the value of the Headmaster property yesterday, and look at the value of the Headmaster property today, and see whether the objects they represent are the same object: whether they are identical. If they are the same object, process 1 has happened, and the headmaster has changed as to one of his properties. If they are not the same object, process 2 has happened, and the headmaster has been replaced. There is identity, or there is non‐identity. There is change, or there is replacement. Our object model allows of no middle way.
Regrettably, our school would never appoint a headmistress. Perhaps it should change as to the value of its Ethos property.
“Changing into”: the language of moot identity
Our object model does not allow of any middle way between identity and non‐identity: between mere change as to the value of an object’s property, and replacement of the object. But, wonderfully, our natural language does. Suppose our school is Hogwarts School of Witchcraft and Wizardry. Then there is a middle way. It is:
- Our headmaster has changed into a man with no beard.
Now that brought you up with a start, because, in the real world, one person does not change into another person. To put it another way, in the real world the identity of a person continues to be clearly recognisable over the whole of his or her lifetime. Indeed a person is perhaps the clearest example of a continuant that there is.
“Changing into”, “turning into”, “growing into”, “being converted into”, “decaying into”, “splitting into”, “merging into”, “evolving into”: these are the language of moot identity. We may add “becoming” in the sense “X becomes Y” where X and Y denote objects. Consider. We say, “A caterpillar changes into a butterfly.” That describes a process of some kind. (I am using the word “butterfly” in its everyday sense, meaning an adult butterfly: if that offends you, substitute “imago” for “butterfly”.) Now ask yourself, Is the identity of the butterfly, after the process, recognisable as the identity of the caterpillar, before the process? Well, the butterfly and the caterpillar are the same organism. There is identity recognisable between them in that zoological sense. But are they the same thing, the same object? In a different sense, no, they are not. The differences between a caterpillar and a butterfly are so profound that we can reasonably choose to say that a caterpillar is one kind of thing, and a butterfly is a different kind of thing. If a toddler pointed at a Red Admiral butterfly flying around on a summer day and exclaimed, “Ook! a capatilla!” then we might marvel at her zoology but we would chuckle at her metaphysics.
If we want to build an object model for this situation, we have a choice. We can treat the caterpillar as one object and the butterfly as another object. Or, we can treat the caterpillar as the same object as the butterfly that it turns into, and give that object a property called “Adult” whose value is False while it is a caterpillar and True when it has become a butterfly. But we have to choose between the two models. We have to decide whether our model will treat the caterpillar before the change and the butterfly after the change as the same object, or not. There is no middle way. Natural language, by contrast, is wonderfully adapted to take care of such cases with its language of moot identity. So where is the greater wisdom: in metaphysics with its black and white, or in natural language with its shades of grey?
Just to confuse you, the verb change can mean “replace”
So far, we have considered the verb change in its intransitive uses only. We have talked of a variable or a property changing (meaning, changing as to its value), and an object changing (meaning, changing as to the value of one or more of its properties). We have looked at the concept of “changing into”, which is another intransitive use of the verb. Turning now to the transitive uses of the verb, we may talk of changing a variable (meaning, changing the variable as to its value), or changing a property of an object (meaning, changing the property as to its value). For example we may talk of changing the shape of a piece of putty, by squeezing it. But when we talk of changing an object we most commonly mean replacing it: think of changing a tyre, or changing a light bulb, or changing a baby’s nappy. In addition the transitive verb has idiomatic uses with different meanings: for example to change a bed means to replace the bedclothes. Natural language is flexible, and can be messy.
Substance and form: a disclaimer
The butterfly example given above is traditionally discussed in terms of substance and form. When a caterpillar changes into a butterfly, there is continuity of identity of the substance (the physical, the concrete, the material) but in a different form (the abstract). Such a change is called a metamorphosis, from Greek words meaning “later form”. But not all cases of “changing into” can be seen in those terms. Here are two very different examples.
You see a chess player moving a pawn forwards from the square a7 towards the square a8. You glance away momentarily, and when you look again, you see the player releasing a knight on a8. What has happened? The chessman has changed as to its size, its shape and its powers? No: that suggests too strongly a continuity of identity. The pawn has been replaced by, or exchanged for, a knight? Well, in physical terms that’s what has happened, but that’s not the way a chess player thinks of it. It suggests too strongly a discontinuity of identity. The pawn has become a knight, or has changed into a knight, or has been promoted into a knight? Yes. These express the right balance between the continuity of identity of a single chessman (the pawn’s future is as a knight, and the knight’s history was as a pawn) and discontinuity of identity (different bit of wood, different kind of chessman).
That example is not like the butterfly example, with continuity of identity of the physical substance in a different abstract form. Largely it is the other way round. There continues to be a chessman of the same colour on the same square, but there is replacement of the physical piece of wood.
In particle physics, there is a process which we describe in natural language as an electron–positron pair “changing into” or “annihilating into” a pair of photons. Now there is no known object whose identity continues through that process, far less any such object that can be said to change its form. As far as we know, the electron and positron just cease to be, and the photons just come to be.
These examples show that when we use a phrase like “changing into” we are by no means always talking about a metamorphosis, that is, a continuity of identity of a physical substance in a different abstract form. Phrases like “changing into” – the language of moot identity – have nothing necessarily to do with questions of substance and form at all.
Consider Sentence A:
- The shy child that I remember has blossomed into the confident man that we see now.
The phrases “the shy child that I remember” and “the confident man that we see now” have the form of referential noun phrases, and such a noun phrase characteristically refers to some entity. What entities, if any, do these noun phrases refer to in this case?
We have to be careful. On any account there is a single human being, a single continuant, underlying both noun phrases. This is not a case of moot identity like the caterpillar and the butterfly. Yet we can’t say that both noun phrases have the same referent: we are not making two references to some person and asserting that he has blossomed into the same person.
The two noun phrases in Sentence A, if they refer to anything at all, refer to entities that have been called object‐stages. Object‐stages have not got a solid reputation among philosophers. Many philosophers would rather do without them. But it doesn’t matter whether you have object‐stages in your ontology, because this essay is about the language of change, and whether or not you posit object‐stages, you have no difficulty in understanding a sentence like Sentence A. Its purported references to object‐stages are a standard manner of speaking. But they are also an avoidable manner of speaking: I could have avoid object‐stages by saying, for example, “I remember him being shy when he was a child, but now that he is a man we can see that he has blossomed and is confident.”
An object‐stage is not a mere property of the underlying object. Those who posit object‐stages contend that an object‐stage is itself a continuant with the same kinds of properties as the underlying object. For example, the noun phrase “the shy child that I remember” refers to a continuant that went to school, played games, liked ice cream, and so on. But it is not the same kind of continuant as a human being, for it ceased to exist when it reached the age of majority. You can see why philosophers are wary of the concept.
If your ontology does not admit of object‐stages, then you have to say that, in Sentence A, the noun phrases “the shy child that I remember” and “the confident man that we see now” do not refer to any entities. But that need not trouble you. Natural language abounds in noun phrases (and proper names, which have the syntactic status of noun phrases) that are referential in form but not referential in meaning. If I say “John F Kennedy was assassinated”, it is my position that the proper name “John F Kennedy” refers to an entity that was assassinated. But if I say “Santa Claus does not exist”, it is not my position that the proper name “Santa Claus” refers to an entity that does not exist. To predicate of an entity that it does not exist would be absurd. Language models the world, but the modelling may be complex.
A case study
I will finish this essay by exploring a sentence that I read some years ago. It shows just how complex may be the concepts underlying an apparently straightforward sentence about change.
Sentence B below was in a news article about Beta Interferon, a drug for the treatment of MS that was quite new at the time:
- This woman has been on Beta Interferon for the last two years and it has made a huge difference to how she was before.
I initially read the phrase “how she was before” as referring to the state of the woman’s health (the value of her Health property) before she took the drug. I thought, What a stupid piece of reporting. How can a drug affect your state of health at a time before you take it?
I showed the sentence to several friends, and I was astonished to find that none of them saw anything wrong with it. When I pointed out to them why I thought it was stupid, a typical reaction was that yes, it might be wrong, but if so the mistake was very subtle and difficult to notice and it wouldn’t have bothered any of them. I was mystified.
It was only when I came to write this essay that I revisited the sentence and realised that there is a way of reading it that makes perfect sense. The phrase “how she was before” need not refer to the state of the woman’s health, as I had read it. It can refer (or purport to refer, depending on your ontology) to an object‐stage of the woman’s health.
Here is the full analysis. The woman is a continuant. Her health is a property of that continuant. Like any property of her, it can be regarded as an object in its own right, and we will so regard it. Next we take two object‐stages of that object: her health during the period before she started taking the drug, and her health during the period after she started taking the drug. The phrase “how she was before” may now be taken as referring to the first of the two object‐stages. And the taking of the drug can indeed be regarded as making a difference to that object‐stage, because it changes that object‐stage into (in the language of moot identity) the second object‐stage. It changes how she was before into how she is now!
Wow. To make analytic sense of the phrase “made a difference to how she was before” we had to start with the woman, take a property called “her health”, regard that property as an object, take two successive object‐stages of that object, and interpret the drug as “making a difference” to the first object‐stage by “changing it into” the second object‐stage in the language of moot identity. What a folk wisdom there is in the natural language of every man, woman and child, that can cut across that analysis and understand the phrase effortlessly at first hearing!
Well, almost every man, woman and child. Evidently it was beyond me.